Exactly soluble random field Ising models in one dimension

Abstract

The authors solve exactly the one-dimensional random field Ising model for two classes of magnetic field distributions: symmetric exponential (model I) and non-symmetric exponential (model II). For both models, expressions for the free-energy at all finite temperatures are presented. The low-temperature region is examined in more detail; they obtain the zero-temperature energy and entropy in closed form; it is shown that the free energy of both models has an expansion in integer powers of temperature. Model I has a non-vanishing zero-point entropy for all values of the parameters as soon as randomness is diluted. In model II the zero-point entropy is zero except for a discrete sequence of values of one parameter. In some cases the zero-temperature magnetisation is positive whereas the average magnetic field is negative; the magnetisation may also change sign as a function of temperature.